Opuscula Math. 41, no. 3 (2021), 335-379
https://doi.org/10.7494/OpMath.2021.41.3.335

Opuscula Mathematica

# Multi-variable quaternionic spectral analysis

Ilwoo Cho
Palle E.T. Jorgensen

Abstract. In this paper, we consider finite dimensional vector spaces $$\mathbb{H}^n$$ over the ring $$\mathbb{H}$$ of all quaternions. In particular, we are interested in certain functions acting on $$\mathbb{H}^n$$, and corresponding functional equations. Our main results show that (i) all quaternions of $$\mathbb{H}$$ are classified by the spectra of their realizations under representation, (ii) all vectors of $$\mathbb{H}^n$$ are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring $$M_n(\mathbb{C})$$ of all $$(n \times n)$$-matrices over the complex numbers $$\mathbb{C}$$ has close connections with certain "non-linear" functional equations on $$\mathbb{H}^n$$ up to the classification of (ii).

Keywords: the quaternions $$\mathbb{H}$$, vector spaces $$\mathbb{H}^n$$ over $$\mathbb{H}$$, $$q$$-spectral forms, $$q$$-spectral functions.

Mathematics Subject Classification: 20G20, 46S10, 47S10.

Full text (pdf)

1. I. Cho, A spectral representation of the quaternions, preprint (2020).
2. I. Cho, P.E.T. Jorgensen, Spectral analysis of equations over quaternions, preprint (2020).
3. C.J.L. Doran, Geometric Algebra for Physicists, Cambridge Univ. Press, 2003.
4. F.O. Farid, Q. Wang, F. Zhang, On the eigenvalues of quaternion matrices, Linear Multilinear Algebra 4 (2011), 451-473.
5. C. Flaut, Eigenvalues and eigenvectors for the quaternion matrices of degree two, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 10 (2002), no. 2, 39-44.
6. P.R. Girard, Einstein’s equations and Clifford algebra, Adv. Appl. Clifford Alg. 9 (1999), no. 2, 225-230.
7. P.R. Halmos, Hilbert Space Problem Book, Springer-Verlag, 1982.
8. P.R. Halmos, Linear Algebra Problem Book, Math. Assoc. Amer., 1995.
9. W.R. Hamilton, Lectures on Quaternions, Cambridge Univ. Press, 1853.
10. I.L. Kantor, A.S. Solodnikov, Hypercomplex Numbers: An Elementary Introuction to Algebras, Springer-Verlag, 1989.
11. V. Kravchenko, Applied Quaternionic Analysis, Heldemann Verlag, 2003.
12. S.D. Leo, G. Scolarici, L. Solombrino, Quaternionic eigenvalue problem, J. Math. Phys. 43 (2002), 5815-5829.
13. N. Mackey, Hamilton and Jacobi meet again: Quaternions and the eigenvalue problem, SIAM J. Matrix Anal. Appl. 16 (1995), no. 2, 421-435.
14. S. Qaisar, L. Zou, Distribution for the standard eigenvalues of quaternion matrices, Internat. Math. Forum 7 (2012), no. 17, 831-838.
15. L. Rodman, Topics in Quaternion Linear Algebra, Prinston Univ. Press, NJ, 2014.
16. B.A. Rozenfeld, The History of non-Eucledean Geometry: Evolution of the Concept of a Geometric Spaces, Springer, 1988.
17. A. Sudbery, Quaternionic Analysis, Math. Proc. Cambridge Philos. Soc. 85 (1998), no. 2, 199-224.
18. L. Taosheng, Eigenvalues and eigenvectors of quaternion matrices, J. Central China Normal Univ. 29 (1995), no. 4, 407-411.
19. J.A. Vince, Geometric Algebra for Computer Graphics, Springer, 2008.
20. J. Voight, Quaternion Algebra, Dept. of Math., Dartmouth Univ., 2020.
• Ilwoo Cho (corresponding author)
• St. Ambrose University, Department of Mathematics and Statistics, 518 W. Locust St., Davenport, Iowa, 52803, USA
• Communicated by P.A. Cojuhari.
• Revised: 2021-02-15.
• Accepted: 2021-02-26.
• Published online: 2021-04-19.